This topic is covered in the book "Dark Side of the Moon: The Magnificent Madness of the American Lunar Quest" by Gerard Degroot.

As I remember it, the author says that Eisenhower was focused on strategic value of investments into space, and he felt that the US had a large lead in areas that mattered (ICBM technology) and chose not to invest in what might be considered vanity projects.

I believe he also claims that letting the Soviets be first to launch a satellite was intentional, since it allowed the Soviets to establish the precedent in international law that it was OK to send satellites over your neighbors, and if the US had been first the USSR might have protested.

As I understand it, the claim is that basically if they let von Braun loose they could have easily beat the Soviets (and he badly wanted to do just that) but one of the major reasons they didn't is that leadership wanted that milestone to be achieved by a U.S. rocket, not a nazi rocket.

https://www.nasa.gov/history/sputnik/begin.html

"Wernher von Braun showed me one of his mock-ups and told me of his plans for orbiting a satellite. He had been unable to obtain the authorizations and funds required to put an actual satellite in orbit; but according to his estimates, he could have done so many months before Russia's Sputnik I was launched-at a minute cost as measured by present-day space expenditures."

"Why was the Redstone-von Braun satellite project not supported? Answers vary with the person talked to: The Navy's brilliant developments in satellite instrumentation had tipped the choice to Vanguard, and budgetary restrictions had prevented a paralleling project. The name Redstone was too closely associated with military missiles. Vanguard offered lower costs, more growth potential, longer duration of orbiting. We would eventually gain more scientific information through Vanguard than through Redstone. To these observations, l can add from my own experience that inter-service rivalry exerted strong influence; also, that any conclusion drawn would be incomplete without taking into account the antagonism still existing toward von Braun and his co-workers because of their service on the German side of World War II."

Where the Soviets were first, the US was typically very close behind (sometimes by just a few days). However, US space missions were of significantly greater value while Soviet missions were often just good enough to claim whatever "first" they were going after.

The claim that one person makes that amount of difference is questionable. Even if you assume he can get the rocket to work reliably. From where they were in 1966, to actually do the landing, and the safe return, that's a whole other thing.